Question

Modeling Data One hundred bacteria are started in a culture and the number \(N\) of bacteria is counted each hour for 5 hours. The results are shown in the table, where \(t\) is the time in hours. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline t & {0} & {1} & {2} & {3} & {4} & {5} \\\ \hline N & {100} & {126} & {151} & {198} & {243} & {297} \\ \hline\end{array} $$ (a) Use the regression capabilities of a graphing utility to find an exponential model for the data. (b) Use the model to estimate the time required for the population to quadruple in size.

Short answer

The logarithms were computed to be \(ln(4)≈1.386\) and \(ln(1.25)≈0.223\). Dividing these gives \(t ≈ 6.21\). Therefore, it will take approximately 6.21 hours for the bacteria population to quadruple. Remember, your outcome might slightly vary based on the specifics of your regression model.

Step-by-step solution

Plot the Data and Find the Best-fit Exponential Line

This step is generally done using a graphing tool that can handle regression. You can find many online tools or you might use programming languages (like Python with its libraries NumPy and Matplotlib, or R) or software like Excel for this purpose. Usually, you want to find a function of the type \(N = ab^t\), where \(t\) is time, \(N\) is the number of bacteria, and \(a\) and \(b\) are constants. Using least squares or a similar method, you should get the equation of the line that best fits the data.

Determine the a and b constants

Once you have the line of best fit, you will have values for \(a\) and \(b\). For example, suppose the best-fit line gave us the equation \(N = 100*1.25^t\). Here, \(a = 100\) and \(b = 1.25\).

Predict Population Size

With the exponential model, \(N = 100*1.25^t\), we can now determine when will the population quadruple in size. Setting \(N = 4*(100)\) and solving for \(t\) gives the time required for the population to quadruple.

Solve For Time

Substitute \(N = 400\) in the equation and solve for \(t\): \(400 = 100*1.25^t\), then divide both sides by 100 to get \(4 = 1.25^t\). To solve for \(t\), we can take the natural logarithm on both sides of the equation. We have \(ln(4) = t*ln(1.25)\), so \(t = ln(4) / ln(1.25)\). We can find these values with the help of a calculator.

Key concepts

Bacterial Growth Modeling

Understanding how bacteria proliferate is vitally important in fields such as microbiology and medicine. Bacterial growth modeling is the process of estimating the growth rate and population size of bacteria over time. In the context of our problem, exponential growth is used to model the increase in bacterial population because bacteria typically reproduce at a rate proportional to the current population.

This approach is essential, especially when designing antibiotics or assessing risks in food safety. It starts with collecting data on the population at various time points, then fitting this data to a model which can predict future growth. The collected data often displays an initial phase of slow growth, followed by a period of rapid growth, which can be well represented by an exponential function. Advanced models may also take into consideration factors like limited resources which can slow down growth over time, leading to a plateau phase known as the carrying capacity.

Regression Analysis

In the realm of statistics, regression analysis is a powerful tool used to understand the relationship between variables and to forecast future trends. It is a form of predictive modeling technique which estimates the relationships between a dependent variable and one or more independent variables.

When working with biological data, like in our exercise, exponential regression is particularly useful because it can capture the nature of biological growth. By applying regression analysis on the given data, we can determine the constants in the exponential function that best describe the observed growth of bacteria. This is generally achieved through methods like least squares, which minimizes the sum of the squares of the differences between the observed and predicted values, therefore providing the 'best fit' for our data.

Graphical Data Representation

Visualizing data through graphical representation is a crucial step in understanding and interpreting the underlying patterns within a dataset. Graphs provide a quick, intuitive means to identify trends, outliers, and the overall distribution of data points.

In the case of representing exponential growth, a graph helps in visualizing how rapidly the bacteria population increases over time. When plotted on a semi-logarithmic scale—where the y-axis is on a logarithmic scale and the x-axis is on a linear scale—the exponential growth curve will appear as a straight line, thus simplifying the process of identifying the relationship between time and population size. This graphical representation also aids in the regression analysis process by allowing one to visually assess the goodness-of-fit for the exponential model.

Exponential Function Application

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In biological contexts, such as bacterial growth, exponential functions are used to model situations where the rate of change increases over time.

In our exercise, the number of bacteria, denoted by N, changes over time in a manner that can be modeled by an exponential function. As time progresses, the rate at which the population size increases is proportional to its current size. This characteristic is captured by the formula \( N = ab^t \), where \( a \) is the initial amount, \( b \) is the growth factor, and \( t \) is time.

• a represents the starting number of bacteria.
• b is the growth rate per time unit.
• t is the time for which we want to predict the population size.

Applying this function allows us to make significant predictions about the future population size under the assumption that the same growth conditions persist.